A treatise on the theory of Bessel functions, by G. N. Watson.

422

THEORY OF BESSEL FUNCTIONS

[CHAP. XIII

To evaluate the integral in the general case, the method discovered by
Hardy is effective; suppose that a > b, and at first let us take R (v)>-,
R (/) >, so that Poisson's integral may be substituted for the second
Bessel function and all the integrals which will be used are absolutely convergent. Write t in place of t + C, and let z - Z= Z, so that the integral to be
evaluated becomes
Jfa(Z+t)} JV(bt)
(Z + t)'~ tv
- ( b) K frJ" J, la (Z +)t)l
= ( ) (~ ) _ Jo (Z +- t) cos (bt cos )) sin'v 0 do dt
(v + ) f ( cos {b (t - Z) cos 0} sin2v 4 do dt
2 ( b1) J t(at) cos (bt cos 4) cos (bZ cos ) sin2v Od4dt
F (7 +) ) r Jo tAo t
= (2)L F( A ( )( I+) (fa - b2 cos2 4)/- cos (bZ cos 4) sin2^" b d,
(2a), r]P (k - ~2 P (V + 2) o
by a special case of ~ 13'4 (2).
This integral is expressible in a simple manner only when A= -, a case
considered by Gallop, or when a = b, the case considered by Hardy.
We easily obtain Gallop's two results
~~(l) In sin a (z Jr t) J (bt) dt = r Jo (bz) (b < a)
(2) J sina(z+ t) d =(b a)
~ _0 z + t o (b) _ =(b u)
and Hardy's formula
0 J a (z + t)} (, + t)} r ( + v) r (+)
2 J2- JL+v-2 (a (z - r)}
The reader will find it interesting to obtain (1) by integrating
f ez +t) Jo (ht) dt
ai(z+t
round the contour formed by the real axis and an indefinitely great semicircle above it; it
has to be supposed that there is an indentation at -z when z is real.
The integral
f: ]t sina(z+t)
i + t Jo (bt) dt
has also been considered by Gallop. To evaluate it, we observe that
t z
z+t - z+t'